In summary, temperature effects lead to regimes in which the minimum of the total axion potential is displaced to \(\theta _a=\pm \, \pi \)  or similar large field values. This high-temperature shift of the minimum parallels that at finite density, previously identified in Refs. [1]}, [2]}, [3]} and recently employed as well in Ref. [4]}.
This phenomenon is rich in consequences. In particular, it follows that the axion mass at finite temperature,
\(m_a(T)=\frac{1}{f_a}\sqrt{\frac{d^2V_{\cal N}(\theta _a,T)}{d\theta _a^2}\Bigg |_{\rm min}}\,\,,\) 
